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SIF AND FINITE ELEMENT SOLUTIONS FOR CORNER SINGULARITIES

  • Woo, Gyungsoo (Department of Mathematics, Changwon National University) ;
  • Kim, Seokchan (Department of Mathematics, Changwon National University)
  • Received : 2018.08.14
  • Accepted : 2018.09.01
  • Published : 2018.09.30

Abstract

In [7, 8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. Their algorithm involves an iteration and the iteration number depends on the acuracy of stress intensity factors, which is usually obtained by extraction formula which use the finite element solutions computed by standard Finite Element Method. In this paper we investigate the dependence of the iteration number on the convergence of stress intensity factors and give a way to reduce the iteration number, together with some numerical experiments.

Keywords

References

  1. H. BLUM AND M. DOBROWOLSKI, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. https://doi.org/10.1007/BF02237995
  2. Z. CAI AND S.C. KIM, A finite element method using singular functions for the poisson equation: Corner singularities, SIAM J. Numer. Anal., 39:(2001), 286-299. https://doi.org/10.1137/S0036142999355945
  3. Z. CAI , S.C. KIM, S.D. KIM, S. KONG, A finite element method using singular functions for Poisson equations: Mixed boundary conditions, Comput. Methods Appl. Mech. Engrg. 195 (2006) 26352648 https://doi.org/10.1016/j.cma.2005.06.004
  4. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
  5. F. HECHT, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3-4, 251265.
  6. S. KIM AND S. KONG, Remarks on finite element methods for corner singularities using SIF, Honam Mathematical J., 38(2016), No.3, 661-674. https://doi.org/10.5831/HMJ.2016.38.3.661
  7. S. KIM AND H.C. LEE, A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor, Computers and Mathematics with Applications, 71(2016) 2330-2337. https://doi.org/10.1016/j.camwa.2015.12.023
  8. S. KIM AND H.-C. LEE, Finite element method to control the domain singularities of Poisson equation using the stress intensity factor : mixed boundary condition, Int. J. Numer. Anal. Model, 14:4-5 (2017), 500-510.